Circles with centers of $(2,2)$ and $(17,10)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?
Explanation: The radius of the first circle is 2, and the radius of the second circle is 10.  The distances between the centers of the circles is $\sqrt{(17 - 2)^2 + (10 - 2)^2} = 17,$ so the distance between the two closest points of the two circle is $17 - 2 - 10 = \boxed{5}.$

[asy]
unitsize(0.3 cm);

draw((2,2)--(2,0),dashed);
draw((17,10)--(17,0),dashed);
draw((-1,0)--(28,0));
draw((0,-1)--(0,20));
draw(Circle((2,2),2));
draw(Circle((17,10),10));
draw((2,2)--(17,10));

label("$2$", (2,1), E);
label("$10$", (17,5), E);

dot("$(2,2)$", (2,2), NW);
dot("$(17,10)$", (17,10), NE);
[/asy]